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<p><dfn class="terminology">Solution:</dfn> Considering the ODE in this form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_15.html">
\begin{equation}
y^{\prime \prime}+p(x) y^{\prime}+q(x) y=g_1(x)+g_2(x).\tag{3.6.5}
\end{equation}
</div>
<p class="continuation">If <span class="process-math">\(y_1\)</span> is a solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_15.html">
\begin{equation*}
y^{\prime \prime}+p(x) y^{\prime}+q(x) y=g_1(x),
\end{equation*}
</div>
<p class="continuation">and <span class="process-math">\(y_2\)</span> is a solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_15.html">
\begin{equation*}
y^{\prime \prime}+p(x) y^{\prime}+q(x) y=g_2(x),
\end{equation*}
</div>
<p class="continuation">then <span class="process-math">\(y_1+y_2\)</span> is a solution of (<a href="" class="xref" data-knowl="./knowl/eq3_15.html" title="Equation 3.6.5">(3.6.5)</a>). For the current problem, we know</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_15.html">
\begin{equation*}
\begin{aligned}
&amp;y^{\prime \prime}-2 y^{\prime}-3 y=3 e^{2 x} \quad \textrm{solution is} ~y_1=-e^{2 x},\\
&amp;y^{\prime \prime}-2 y^{\prime}-3 y=2 \sin x \quad  \textrm{solution is}~y_2=-\frac{2}{5} \sin x+\frac{1}{5} \cos x,\\
&amp;y^{\prime \prime}-2 y^{\prime}-3 y=2 x^2 \quad  \textrm{solution is}~y_3=-\frac{2}{3}x^2+\frac{8}{9}x-\frac{28}{27},\\
&amp;y^{\prime \prime}-2 y^{\prime}-3 y=2 e^x \sin x \quad  \textrm{solution is}~y_4=-\frac{2}{5} e^x \sin x.\\
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus, the solution for our problem is (note the superposition)</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_15.html">
\begin{equation*}
y=y_1+y_2+y_3+y_4=-e^{2 x}-\frac{2}{5} \sin x+\frac{1}{5} \cos x-\frac{2}{3}x^2+\frac{8}{9}x-\frac{28}{27}-\frac{2}{5} e^x \sin x.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_6.html#p-128" class="internal">in-context</a></span>
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